# Near-Optimal Sample Complexity Bounds for Maximum Likelihood Estimation of Multivariate Log-concave Densities

arXiv: https://arxiv.org/abs/1802.10575 · HTML: https://mlsanity.com/paper/1802.10575/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Feb 2018.

## Built on by (most cited first, 3 of 3)

- [On the Variance, Admissibility, and Stability of Empirical Risk Minimization](https://mlsanity.com/paper/2305.18508/) (2023)
- [An Efficient Minimax Optimal Estimator For Multivariate Convex Regression](https://mlsanity.com/paper/2205.03368/) (2022)
- [Density estimation for Hellinger via minimum-distance estimators: mixtures of Gaussians, log-concave, and more](https://mlsanity.com/paper/2606.11469/) (2026)

## Abstract (arXiv)

We study the problem of learning multivariate log-concave densities with respect to a global loss function. We obtain the first upper bound on the sample complexity of the maximum likelihood estimator (MLE) for a log-concave density on ℝd, for all d ≥ 4. Prior to this work, no finite sample upper bound was known for this estimator in more than 3 dimensions. In more detail, we prove that for any d ≥ 1 and ε>0, given tildeOd((1/ε)(d+3)/2) samples drawn from an unknown log-concave density f0 on ℝd, the MLE outputs a hypothesis h that with high probability is ε-close to f0, in squared Hellinger loss. A sample complexity lower bound of Omegad((1/ε)(d+1)/2) was previously known for any learning algorithm that achieves this guarantee. We thus establish that the sample complexity of the log-concave MLE is near-optimal, up to an tildeO(1/ε) factor.